For a corresponding lower bound, we note that Lemma 1 in [1]} yields that
\(\binom{N}{k}-{\mathbb {E}}\,f_k(C_N)&=\binom{N}{k}\frac{A}{1+A}\\&=\binom{N}{k}\frac{1}{2^{N-1} P_{d,N}}\sum _{j=1}^{k}\binom{k}{j}\sum _{m=0}^{j-1}\binom{N-k-1}{d-k+m}.\) 
